advertisement

Homework 3 Problem 1 a) Assume that the temperature is circularly symmetric: u = u(r, t), where r 2 = x2 + y 2. ∂ = kr ∂r r ∂u Derive the heat equation in a circular annulus a ≤ r ≤ b without the source, ∂u ∂t ∂r by writing the fluxes and rate of energy change. b) Determine the equilibrium temperature if the temperature of the outer circle is T1 and the inner circle is T2 . c) Repeat part a) and part b) for two concentric spheres with radii a and b. Problem 2 For conduction of thermal energy, the heat flux is φ = −K0 ∇u. If in addition the molecules move at an average velocity v, a process is called convection, then briefly explain why φ = −K0 ∇u + cρuv. Derive the corresponding equation for heat flow, including both conduction and convection of thermal energy (assume no source). Problem 3 Using MATLAB solve the heat conduction problem ∂u = ∇ · (k(x)∇u) ∂t in an inhomogeneous bar. Assume the bar has fixed temperature T = 1 at the ends, and initial temperature T = 0. a) Assume k = 0.001 if x < 0.6 and k = 1 elsewhere. b) Assume k = 1 if 0.3 < x < 0.7 and k = 0.001 elsewhere. c) Assume k = 0.001 everywhere. Solve the problem till tf inal = 2. Answer the following questions. What is the difference between a) and b)? Why? What is the difference between b) and c)? Why? 1